Spin-polarization in quantum wires: Influence of Dresselhaus spin-orbit interaction and crosssection

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1 University of Richmond UR Scholarship Repository Physics Faculty Publications Physics Spin-polarization in quantum wires: Influence of Dresselhaus spin-orbit interaction and crosssection effects L. Villegas-Lelovsky C. Trallero-Giner Mariama Rebello Sousa Dias University of Richmond, V. Lopez-Richard G. E. Marques Follow this and additional works at: Part of the Physics Commons Recommended Citation Villegas-Lelovsky, L., C. Trallero-Giner, M. Rebello Sousa Dias, V. Lopez-Richard, G.E. Marques. "Spin-polarization in quantum wires: Influence of Dresselhaus spin-orbit interaction and cross-section effects." Physical Review B 79, no.15 (April 2009): This Article is brought to you for free and open access by the Physics at UR Scholarship Repository. It has been accepted for inclusion in Physics Faculty Publications by an authorized administrator of UR Scholarship Repository. For more information, please contact scholarshiprepository@richmond.edu.

2 PHYSICAL REVIEW B 79, Spin polarization in quantum wires: Influence of Dresselhaus spin-orbit interaction and cross-section effects L. Villegas-Lelovsky, 1,2 C. Trallero-Giner, 3 M. Rebello Sousa Dias, 2 V. Lopez-Richard, 2 and G. E. Marques 2 1 Departamento de Ciências Físicas, Universidade Federal de Uberlândia, Uberlândia, Minas Gerais, Brazil 2 Departamento de Física, Universidade Federal de São Carlos, São Carlos, São Paulo, Brazil 3 Departamento de Física Teórica, Universidad de La Habana, Havana, Cuba Received 6 November 2008; revised manuscript received 14 February 2009; published 8 April 2009 We examine the effects of the full Dresselhaus spin-orbit coupling on laterally confined quantum wire states. An analysis of the relative contributions due to linear, quadratic, and cubic Dresselhaus spin-orbit terms on the energy levels, spin splitting, and spin polarization has been carried out. The effects of wire cross-sectional geometry shapes on the electronic structure are explored. In particular we compared the results of semicylindrical and cylindrical confinements and have found important differences between the spin degeneracy of the ground-state level and the spin-polarization dependence on sign inversion of the free linear momentum quantum number along the wire axis. Different from other symmetries, in both cases here considered, the stronger spin-splitting effects come from the quadratic Dresselhaus term. We report ideal conditions for realization of spin-field filter devices based on symmetry properties of the spin splitting of the ground state in semicylindrical quantum wires. DOI: /PhysRevB PACS number s : Hb, Ej, Dj, Dc I. INTRODUCTION Bulk spin-orbit interaction SOI effects of zinc-blende crystals have been discussed in earlier theoretical papers. 1 3 They break the spin degeneracy of the Bloch states for every wave vector k in the absence of a center of inversion and lead to the energy spin splitting of the electron and/or hole states without lifting Kramer s degeneracy in the absence of an external magnetic field. In low dimensional heterostructures, this spin splitting can be tuned by external field and classified into two effects either induced by bulk inversion asymmetry BIA in zinc-blende crystal structures, known as Dresselhaus SOI, and/or by structure inversion asymmetry SIA of the confinement potential. The intrinsic form of BIA SOI which affects the energy level separation of carrier states in zinc-blende nanostructures can be detected via spin-polarized optical and transport measurements. Recently, the BIA-induced spin splitting has been detected experimentally through the Shubnikov de Haas effect in uniaxially strained bulk InSb and by analyzing the precession of the spin polarization of photoexcited electrons along the 110 GaAs surfaces. 4 In turn, SIA-induced spin splitting has been detected via analysis of optical polarized emissions from electron and hole charge densities accumulated in layers of GaAs-AlAs tunneling diodes 5 under applied parallel electric and magnetic fields. In this work we present a detailed k p analysis of the electronic structure in semiconductor quantum wires QWRs of different geometrical shapes after including all Dresselhaus SOI terms in the Hamiltonian for the conduction band states which contain the full symmetry of the crystal. 3,6,7 QWRs have attracted attention because of their potential applications in spintronic and optoelectronic devices. The study of these systems allows us to understand the role played by dimensionality, 8 by electron-electron interaction, 9 as well as by size and shape quantization effects on their electronic properties. 10,11 In general, the Dresselhaus SOI coupling has been introduced to the electronic structure of nanostructures by taking the mean values of the wave-vector operators kˆz 2 and kˆz, as usually done in quantum dots 12 QD s or in a twodimensional 2D electron gas in quantum wells QW s. 4 In most cases, kˆz =0 and such approaches consider only effects due to in-plane kˆ-linear terms proportional to the inplane k components k x,k y. 3,8,13,14 For other types of confinements, as well as for materials with a large values of SOI parameters, the kˆ-cubic term must be included in the total Hamiltonian. Recently, the effect of the k-cubic term has appeared to be relevant even for GaAs-AlGaAs, a system with very small SOI parameters. 15,16 Moreover and due to the translational invariance symmetry of one-dimensional 1D quantum systems along the z axis, the BIA quadratic term, proportional to kˆz, should also have a relevant contribution to the electronic subband structure of QWRs. Later on we shall discuss why SOI quadratic and cubic terms are essential to determine eigenenergies, spin-splitting energies, and spin polarizations of electrons in the conduction band of QWRs. By changing the ratio between SOI and quantum confinement strengths we can follow the relative influences of each BIA term on the electronic spectra. Furthermore, besides the qualitative differences between circular and semicircular cross-section QWRs, we will also show that any symmetry reduction leads to a set of eigenstates affected by different combinations of SOI linear, quadratic, and cubic Dresselhaus contributions. Hence, symmetry considerations to determine the Hilbert space for the solutions used in the expansion of the spinor states and the relative effects between different SOI terms become the main issue under discussion in this work. In this study, we will focus on the interplay between linear, quadratic, and cubic BIA contributions to the spin-splitting energy and the corresponding spin polarization of the ground state of nanoscopic QWRs. In our analysis we have found that the kˆ-linear and kˆ-cubic terms keep the degeneracy of the eigenenergies in a semicylindrical QWR and, in consequence, their contribution to the spinsplitting energy is negligible. Finally, we addressed the ef /2009/79 15 / The American Physical Society

3 VILLEGAS-LELOVSKY et al. PHYSICAL REVIEW B 79, FIG. 1. Quantum wire systems with semicylindrical and cylindrical geometric cross sections. fects associated to the reduction in the cylindrical symmetry on the dependence of the ground-state energy on the sign of wave vector k z along the propagation direction and its notably high degree of spin polarization at low velocity values. II. INFINITE QUANTUM WIRE SYSTEMS The total single-particle Hamiltonian for the conduction band of a QWR, including all Dresselhaus SOI contributions, can be written as H = H 0 + H D, where H 0 = 2 kˆ 2 /2m +V r,z I is the Hamiltonian for uncoupled spin-up and spin-down electron states, I is the 2 2 identity matrix, V r,z is the total spatial confinement for Cartesian coordinates r= x,y and z along the wire axis see Fig. 1, H D = ˆ is the BIA Hamiltonian, m and are the electron effective mass and Dresselhaus SOI parameter for the material, respectively, = x, y, z is the Pauli spin-vector matrix, and ˆ is a vector operator with ˆ x = kˆykˆxkˆy kˆzkˆxkˆz and cyclic permutations for the other ˆ y and ˆ z components and of the associated linear momentum operator kˆ = kˆx,kˆy,kˆz. The effective full BIA Hamiltonian can be separated into three contributions with the form H D = E 0 a 3 H 2D H 1D + H 3D 1 H 1D + H 3D, 2 H 2D where H 1D = D kˆz 2 kˆ, H 2D = 1 2 Dkˆz kˆ 2 +kˆ+ 2, H 3D = 1 8 D kˆ+, kˆ+ 2 kˆ 2, and Â,Bˆ = 1 ÂBˆ 2 +Bˆ Â. These terms are, respectively, the linear, the quadratic, and the cubic Dresselhaus SOI contributions written in terms of operators kˆ =kˆx ikˆy multiplied by the dimensionless parameter D = / a 3 E 0, where E 0 = 2 / 2m a 2 is the energy scale for the QWR confinement inside a cylinder or semicylinder of radius a see Fig. 1. For free motion along the z axis V r,z V r, we can show that the commutator kˆz,h =0 thus, the eigenvalue of kˆz operator k z becomes a good quantum number. Under these conditions, the Schrödinger equation for H is separable, in the entire level spectrum, as a product of purely in-plane localized function on the radial coordinate r times free motion function along z axis. For mathematical simplicity we will assume a hard-wall confining potential model with V r 0 inside the region r a and infinite barrier outside the wire with V r a =. Without SOI, the eigenenergies of the Hamiltonian in Eq. 1 become degenerate and correspond to the spinor eigenfunctions, + f and 0 f 0, where and + are the electron spin-up and spin-down eigenstates of the z component of the spin operator, Ŝ z. The ket f 0 can be factorized as r,z f 0 = f 0 r,z =f r exp ik z z, where f r is the solution of in-plane Hamiltonian equation, 2 2 2m k z 2m f = E 0 f, where 2 is the Laplace operator expressed in polar coordinates, r=,, and E 0 is the eigenvalue of the Hamiltonian H 0 in Eq. 1. The corresponding boundary conditions for a cylinder with circular cross section right panel of Fig. 1 are given by f a, =0, f, = f, +2, 4 and for semicircular cross section left panel of Fig. 1 by f a, =0, f,0 =0, f, =0. 5 In order to fulfill conditions 4 or 5, the eigensolutions, f n,m,, with quantum numbers n and m are products of integer-order Bessel functions, J n u, times the corresponding circular eigenfunctions, 3 c n,m J n n,m a e f n,m = A in, n =0, 1, 2,. 6 A sc n,m J n n,m n =1,2,... a sin n, In these expressions, n,m = E0 /E 0 2 z are the mth zero of the nth-order Bessel function, J n n,m =0, and z =ak z. The superscripts, c and sc, label the solutions corresponding to cylindrical and semicylindrical confinements with normalization constants A n,m =1/ a Jn+1 n,m and A n,m c sc =2/ a Jn+1 n,m, respectively. The eigenfunctions of the total Hamiltonian H, ineq. 1, are represented by spinors + +, where the offdiagonal contributions mix spin-up and spin-down compo- nents of Ŝ z. They are determined by the 2 2 Schrödinger equation, H 0 k z + H 2D k z E H 1D k z + H 3D k z H 1D k z + H 3D k z H 0 k z H 2D k z E + + =

4 SPIN POLARIZATION IN QUANTUM WIRES: INFLUENCE It is straightforward to show that the functions defined in Eq. 6 form a complete set of orthonormal eigenstates for the spaces determined by Hamiltonian 7 with boundary conditions 4 or 5. Therefore, we can build each component of the spinor as a linear combination of functions given by Eq. 6,, = C n,m f n,m,. 8 n,m In principle, the above expansion yields infinite generalized eigenvalue problems for the Hilbert space of Eq. 7. Itbecomes clear that each Dresselhaus contribution appearing in H causes different admixtures between the quantum numbers PHYSICAL REVIEW B 79, n and m of the spin-up and spin-down components of spinor states. As will be shown in Secs. III A and III B, the semicylindrical QWR symmetry imposes restrictive conditions on the spin-degenerate energy levels and on the corresponding spin-carrier polarization Ŝ z of the ground state. III. ENERGY LEVELS AND SPIN CARRIER POLARIZATION We can search for solutions of by direct substitution of expansion 8 into Eq. 7. Then, we end up having to solve an eigenvalue problem, H IE C=0, where H = H is an l l -square matrix the l value accounts for the convergence of the numerical procedure to solve the secular equation with elements given by H = s z 1 2 D z h D 1 8 j + D 1 8 j z g + 2 z g 2 s + 2 z D z h 9 and C= C+ is a vector that forms the set of eigenstates of C the matrix in Eq. 9. The index s represents the set of quantum numbers n,m, ordered in increasing values of n,m. The innermost matrix elements g =a s kˆ s, h =a 2 s kˆ+ 2 +kˆ 2 s, and j =a 3 s kˆ 2, kˆ+ 2 kˆ 2 s can be calculated as shown in the Appendix. In order to simplify the identification of each state admixture induced by the SOI, let us label the components of a spinor eigenstate as n,m; according to the character of the state at zero BIA interaction given by pure or uncoupled =+ or = spin states with eigenvalues /2 along the quantization z axis, respectively. In the sequence, we will be discussing the mixing effects that affect energy levels, spinsplitting energy, and spin polarization caused by the linear, quadratic, cubic, and full Dresselhaus SOI terms for wires with cross sections shown in Fig. 1. It is observed, in Eq. 9, that under the inversion transformation, k z k z, we obtain the same conduction subband mirror states by only changing spin direction. Hence, in Sec. III A we present the electronic spectrum for k z 0 only. A. Cylindrical cross-section confinement In the absence of SOI the eigenstates of Eq. 3 for cylindrical QWRs and with boundary conditions given in Eq. 4 have eigenenergies E n,m =E 0 E z =E 0 n,m. These states labeled n,m; are twofold degenerate for the z component of the angular momentum n=0 and fourfold degenerate for n 0. The mixing induced by BIA terms in H D breaks the spin degeneracy of QWR states in different manners. Following the calculation given in the Appendix we observe that the matrix elements of the off-diagonal linear, g, and cubic, j, Dresselhaus terms see Eqs. A2 and A5 couple states with opposite spins interspin coupling with n=n n= 1 and n= 3, 1, respectively. However, the quadratic BIA spin-orbit coupling occurring in the diagonal of Eq. 9, h, connects those states in Eq. 8 with n= 2 see Eq. A3 but inside the same spinor component intraspin coupling preserving the spin polarization between coupled quantum numbers n,m. The energy splitting induced by this quadratic coupling is similar to the Zeeman splitting induced by a magnetic field; besides, it does not mix different n-index parities of the Bessel functions entering the linear combination of Eq. 8. According to the present selection rules, the Hilbert space for the spinors r can be split into two independent subspaces, labeled U and L, which are classified according to the parity of the quantum number n and spin orientation forming the components of each spinor. Thus, the Hilbert subspace U L gathers spinor states with spin-up spin-down components having odd n values even n values that are coupled with states with spin-down spin-up and even n values odd n values. Hence, the eigenvalue problem for the Hamiltonian in Eq. 7 can be solved independently for each class of states L and U. In Fig. 2 a we are showing effects caused by the BIA terms on the first 12 eigenvalues of H 0 as a function of the normalized Dresselhaus parameter D = / a 3 E 0 and for a giving value of the dimensionless wave number z =k z a, forming the U class eigenstates. Here, the fourfold degener

5

6

7

8 SPIN POLARIZATION IN QUANTUM WIRES: INFLUENCE m = 1, whereas the semicylindrical QWR symmetry, fulfilling the boundary conditions in Eq. 5, forbids the presence of states with n 0. Therefore, under this geometry neither H 1D nor H 3D contributes to the energy spin splitting besides leaving the states twofold degenerate. As seen in Fig. 5 b, their effects produce only small shifts to the energy levels. Finally, the spin polarization of ground states of a semicylindrical QWR, z 1,1, is qualitatively different from the cylindrical case due to different contributions induced by Dresselhaus SOI. For any level, there is not spin-degeneracy for a fixed value of k z in contrast to the cylindrical case where there are always twofold spin-degenerate levels at any fixed value of k z, regardless its propagation direction z. The semicylindrical confinement has given rise to an energy splitting of twofold degenerate levels of QWRs produced by the H 2D term, a dominant process that preserves spin polarization at small values of k z. This opens the possibility of using low velocity v z = k z /m transport measurements to explore preferential spin channels according to the sign propagation direction of induced spin-polarized currents in these quasi-1d nanostructures. Moreover, using this peculiar symmetry for the ground state in semicylindrical QWRs we can expect the formation of robust spin-filter devices. If electrons with energy E/E 0 17 are injected along the z axis of semicylindrical ensemble of parallel QWRs, this device can only transmit a high degree of spin-up current in direction +z and opposite spins in the z direction. A dominant spin-up or spin-down character of the current depends on the sign of the z, and the current density is directly linked to the interplay between the radius a and the wave vector k z, as seen in Fig. 7. The ratio 2m / 2 =5 Å, used in Fig. 7, corresponds to totally realistic values for InAs semiconductors. 18 For cylindrical radius 30 a 60 Å and 0 k z 0.5 Å 1, it is possible to get more than 90% degree of spin-polarized currents in the independent spin-up and spin-down channels. These results are based solely on the spatial symmetry properties of the quasi-one-dimensional nanostructures and enable realization of spin filters and tuned spin-polarized current densities in both parallel QWR directions. Also, combination of parallel wires may be used to generate logical gates, necessary for quantum information transmission and spintronic devices. It is worth commenting on another SOI contribution provoked by the structural inversion asymmetry in nanostructure, the so-called Rashba interaction. The Rashba spinsplitting Hamiltonian, H R = k V, 19 in a material with coupling parameter is associated to spatial asymmetries of the lateral confinement potential, V x,y. This contribution is linear on k components and presents the same mathematical structure as the H 1D term. It is clear that eigenstates of H 0 +H R are also eigenstates of H 0 +H 1D. The effect of this term adds to the linear Dresselhaus SOI, for more detailed discussion see Ref. 11. For fixed SO-coupling constant, the spin splitting induced by H 1D is positive see Figs. 3 and 6, 1 whereas for H R the splitting n,m has negative values. Therefore, the net effect of H 1D +H R is to minimize the contribution from H 1D and decrease the values of 1 n,m. 11 Moreover, the efficiency of the proposed spin-filter devices should decline for high impurity concentration or under strong current density injection where the electron-electron interaction becomes important. The validity of our proposition is directly linked to the conservation of the wave vector k z. In this sense, these two effects break translational invariance along the z axis and the single-particle problem based on k z conservation would require another formulation. ACKNOWLEDGMENTS This work was partially supported by CONACYT/Mexico and Brazilian agencies FAPESP, CNPq, and FAPEMIG. L.V-L wants to thank E. Gomez and H. Tsuzuki for technical assistance. G.E.M. and C.T-G. are grateful for the Visiting Scholar Program of the ICTP/Trieste. APPENDIX: SPIN ORBIT TERMS The matrix elements g, h, and j are necessary in order to build the Hamiltonian matrix 9 for the BIA SOI. In polar coordinates the operators kˆ are written as kˆ = ie i i. 1. Cylindrical confinement A1 Using the wave functions Eq. 6 it is straightforward to show that g = a s kˆ s = 2iT n,m n,m n,n 1, A2 where s= n,m and s = n,m, with h = a 2 s kˆ 2 + kˆ+ 2 s =4 n +1 n,n+2 n 1 n,n 2 T n,m n, A3,m n,m 2 lim +1 T n 1 n n 2,m =2 n,m m,m, and for the cubic contribution we have j where = a 3 s kˆ 2, kˆ+ 2 kˆ 2 s = it n,m n,m n 1 n 2 n m n,n 3, T n,m n,m / n,m n =,m n,m / n,m Semicylindrical confinement 2 2 n m n,n 1 A4 A5 A6 Following Eq. 6 for the semicylindrical case we have that + g PHYSICAL REVIEW B 79, n,m = i sgn n n T n 1,m, if n = n 1, A

9 VILLEGAS-LELOVSKY et al. + g = 2n 1+ 1 n+n s J n +1 s J n +1 s F n,m n,m, if n n 1, A8 s where sgn x =+1ifx 0 and sgn x = 1 if x 0 and F s are numbers ruled by s F s s = 0 zjn J s s z n 1 z 1 n 2 n 2 J n +1 z 1+n 2 n 2 dz. A9 In the particular case where s=s Eq. A9 can be reduced to PHYSICAL REVIEW B 79, F s 2n s = s z d 2n dz J n z 2 + J n z 2 dz 2n d = 2n dz z J n z 2 dz =0. Also, it follows the relation s A10 g = 1 n +n+1 g +. A11 The other matrix elements for the semicylindrical QWR, h and j, are evaluated numerically using Eqs. A7 A11 and the matrix identity s ÂBˆ s = p s  p p Bˆ s. 1 R. H. Parmenter, Phys. Rev. 100, G. Dresselhaus, Phys. Rev. 100, A detailed analysis, including all three Dresselhaus contributions for the twofold lowest subbands in zinc-blende bulk structures, has been carried out in G. E. Marques, A. C. R. Bittencourt, C. F. Destefani, and S. E. Ulloa, Phys. Rev. B 72, R. Winkler, Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems, Springer Tracts in Modern Physics Springer, Berlin, 2003, Vol. 191, and references therein. 5 H. B. de Carvalho, M. J. S. P. Brasil, V. Lopez-Richard, Y. Galvao Gobato, G. E. Marques, I. Camps, L. C. O. Dacal, M. Henini, L. Eaves, and G. Hill, Phys. Rev. B 74, R 2006 ; see also L. F. dos Santos, Y. G. Gobato, V. Lopez-Richard, G. E. Marques, M. J. S. P. Brasil, M. Henini, and R. J. Airey, Appl. Phys. Lett. 92, S. Gopalan, J. K. Furdyna, and S. Rodriguez, Phys. Rev. B 32, F. G. Pikus and G. E. Pikus, Phys. Rev. B 51, J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, ; S. Bandyopadhyay, S. Pramanika, and M. Cahay, Superlattices Microstruct. 35, ; S. Kettemann, Phys. Rev. Lett. 98, G. Fasol and H. Sakaki, Phys. Rev. Lett. 70, F. M. Alves, G. E. Marques, V. Lopez-Richard, and C. Trallero- Giner, Semicond. Sci. Technol. 22, F. M. Alves, C. Trallero-Giner, V. Lopez-Richard, and G. E. Marques, Phys. Rev. B 77, C. F. Destefani, S. E. Ulloa, and G. E. Marques, Phys. Rev. B 69, F. Malet, M. Pi, M. Barranco, L. Serra, and E. Lipparini, Phys. Rev. B 76, S. Pramanik, S. Bandyopadhyay, and M. Cahay, Phys. Rev. B 68, D. M. Zumbühl, J. B. Miller, C. M. Marcus, D. Goldhaber- Gordon, J. S. Harris, Jr., K. Campman, and A. C. Gossard, Phys. Rev. B 72, R J. J. Krich and B. I. Halperin, Phys. Rev. Lett. 98, These results are independent of the lateral confining potential used to describe the in-plane localization in cylindrical QWRs. If another two-dimensional confining potential is employed instead of the hard-wall model, the stronger spin-splitting effect still comes from the quadratic Dresselhaus term since the translational symmetry is preserved along wire axis. 18 Parameters for InAs semiconductor m /m 0 = from Landolt-Börnstein Comprehensive Index, edited by O. Madelung and W. Martienssen Springer, Berlin, 1996 and =27.18 ev Å 3 from Ref Yu. A. Bychkov and E. I. Rashba, Sov. Phys. JETP 39, ; J. Phys. C 17,